Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent

This paper considers the problem of recovering a tensor with an underlying low-tubal-rank structure from a small number of corrupted linear measurements. Traditional approaches tackling such a problem require the computation of tensor Singular Value Decomposition (t-SVD), that is a computationally intensive process, rendering them impractical for dealing with large-scale tensors. Aim to address this challenge, we propose an efficient and effective low-tubal-rank tensor recovery method based on a factorization procedure akin to the Burer-Monteiro (BM) method. Precisely, our fundamental approach involves decomposing a large tensor into two smaller factor tensors, followed by solving the problem through factorized gradient descent (FGD). This strategy eliminates the need for t-SVD computation, thereby reducing computational costs and storage requirements. We provide rigorous theoretical analysis to ensure the convergence of FGD under both noise-free and noisy situations. Additionally, it is worth noting that our method does not require the precise estimation of the tensor tubal-rank. Even in cases where the tubal-rank is slightly overestimated, our approach continues to demonstrate robust performance. A series of experiments have been carried out to demonstrate that, as compared to other popular ones, our approach exhibits superior performance in multiple scenarios, in terms of the faster computational speed and the smaller convergence error.

SVDinsTN: An Integrated Method for Tensor Network Representation with Efficient Structure Search

Tensor network (TN) representation is a powerful technique for data analysis and machine learning. It practically involves a challenging TN structure search (TN-SS) problem, which aims to search for the optimal structure to achieve a compact representation. Existing TN-SS methods mainly adopt a bi-level optimization method that leads to excessive computational costs due to repeated structure evaluations. To address this issue, we propose an efficient integrated (single-level) method named SVD-inspired TN decomposition (SVDinsTN), eliminating the need for repeated tedious structure evaluation. By inserting a diagonal factor for each edge of the fully-connected TN, we calculate TN cores and diagonal factors simultaneously, with factor sparsity revealing the most compact TN structure. Experimental results on real-world data demonstrate that SVDinsTN achieves approximately $10^2\sim{}10^3$ times acceleration in runtime compared to the existing TN-SS methods while maintaining a comparable level of representation ability.

Fast and Structured Block-Term Tensor Decomposition For Hyperspectral Unmixing

The block-term tensor decomposition model with multilinear rank-$(L_r,L_r,1)$ terms (or, the "LL1 tensor decomposition" in short) offers a valuable alternative for hyperspectral unmixing (HU) under the linear mixture model. Particularly, the LL1 decomposition ensures the endmember/abundance identifiability in scenarios where such guarantees are not supported by the classic matrix factorization (MF) approaches. However, existing LL1-based HU algorithms use a three-factor parameterization of the tensor (i.e., the hyperspectral image cube), which leads to a number of challenges including high per-iteration complexity, slow convergence, and difficulties in incorporating structural prior information. This work puts forth an LL1 tensor decomposition-based HU algorithm that uses a constrained two-factor re-parameterization of the tensor data. As a consequence, a two-block alternating gradient projection (GP)-based LL1 algorithm is proposed for HU. With carefully designed projection solvers, the GP algorithm enjoys a relatively low per-iteration complexity. Like in MF-based HU, the factors under our parameterization correspond to the endmembers and abundances. Thus, the proposed framework is natural to incorporate physics-motivated priors that arise in HU. The proposed algorithm often attains orders-of-magnitude speedup and substantial HU performance gains compared to the existing three-factor parameterization-based HU algorithms.

Nonstationary Temporal Matrix Factorization for Multivariate Time Series Forecasting

Modern time series datasets are often high-dimensional, incomplete/sparse, and nonstationary. These properties hinder the development of scalable and efficient solutions for time series forecasting and analysis. To address these challenges, we propose a Nonstationary Temporal Matrix Factorization (NoTMF) model, in which matrix factorization is used to reconstruct the whole time series matrix and vector autoregressive (VAR) process is imposed on a properly differenced copy of the temporal factor matrix. This approach not only preserves the low-rank property of the data but also offers consistent temporal dynamics. The learning process of NoTMF involves the optimization of two factor matrices and a collection of VAR coefficient matrices. To efficiently solve the optimization problem, we derive an alternating minimization framework, in which subproblems are solved using conjugate gradient and least squares methods. In particular, the use of conjugate gradient method offers an efficient routine and allows us to apply NoTMF on large-scale problems. Through extensive experiments on Uber movement speed dataset, we demonstrate the superior accuracy and effectiveness of NoTMF over other baseline models. Our results also confirm the importance of addressing the nonstationarity of real-world time series data such as spatiotemporal traffic flow/speed.

Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion

The linear transform-based tensor nuclear norm (TNN) methods have recently obtained promising results for tensor completion. The main idea of this type of methods is exploiting the low-rank structure of frontal slices of the targeted tensor under the linear transform along the third mode. However, the low-rankness of frontal slices is not significant under linear transforms family. To better pursue the low-rank approximation, we propose a nonlinear transform-based TNN (NTTNN). More concretely, the proposed nonlinear transform is a composite transform consisting of the linear semi-orthogonal transform along the third mode and the element-wise nonlinear transform on frontal slices of the tensor under the linear semi-orthogonal transform, which are indispensable and complementary in the composite transform to fully exploit the underlying low-rankness. Based on the suggested low-rankness metric, i.e., NTTNN, we propose a low-rank tensor completion (LRTC) model. To tackle the resulting nonlinear and nonconvex optimization model, we elaborately design the proximal alternating minimization (PAM) algorithm and establish the theoretical convergence guarantee of the PAM algorithm. Extensive experimental results on hyperspectral images, multispectral images, and videos show that the our method outperforms linear transform-based state-of-the-art LRTC methods qualitatively and quantitatively.

Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

The robust tensor completion (RTC) problem, which aims to reconstruct a low-rank tensor from partially observed tensor contaminated by a sparse tensor, has received increasing attention. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{c}$onvex optimization model (RC-FCTN) for the RTC problem. Then, we rigorously establish the exact recovery guarantee for the RC-FCTN. For solving the constrained optimization model RC-FCTN, we develop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. Moreover, we suggest a $\textbf{FCTN}$-based $\textbf{r}$obust $\textbf{n}$on$\textbf{c}$onvex optimization model (RNC-FCTN) for the RTC problem. A proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we theoretically derive the convergence of the PAM-based algorithm. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that proposed methods are superior to several state-of-the-art methods.

Nonlocal Patch-Based Fully-Connected Tensor Network Decomposition for Remote Sensing Image Inpainting

Remote sensing image (RSI) inpainting plays an important role in real applications. Recently, fully-connected tensor network (FCTN) decomposition has been shown the remarkable ability to fully characterize the global correlation. Considering the global correlation and the nonlocal self-similarity (NSS) of RSIs, this paper introduces the FCTN decomposition to the whole RSI and its NSS groups, and proposes a novel nonlocal patch-based FCTN (NL-FCTN) decomposition for RSI inpainting. Different from other nonlocal patch-based methods, the NL-FCTN decomposition-based method, which increases tensor order by stacking similar small-sized patches to NSS groups, cleverly leverages the remarkable ability of FCTN decomposition to deal with higher-order tensors. Besides, we propose an efficient proximal alternating minimization-based algorithm to solve the proposed NL-FCTN decomposition-based model with a theoretical convergence guarantee. Extensive experiments on RSIs demonstrate that the proposed method achieves the state-of-the-art inpainting performance in all compared methods.

Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery

In this paper, we study multi-dimensional image recovery. Recently, transform-based tensor nuclear norm minimization methods are considered to capture low-rank tensor structures to recover third-order tensors in multi-dimensional image processing applications. The main characteristic of such methods is to perform the linear transform along the third mode of third-order tensors, and then compute tensor nuclear norm minimization on the transformed tensor so that the underlying low-rank tensors can be recovered. The main aim of this paper is to propose a nonlinear multilayer neural network to learn a nonlinear transform via the observed tensor data under self-supervision. The proposed network makes use of low-rank representation of transformed tensors and data-fitting between the observed tensor and the reconstructed tensor to construct the nonlinear transformation. Extensive experimental results on tensor completion, background subtraction, robust tensor completion, and snapshot compressive imaging are presented to demonstrate that the performance of the proposed method is better than that of state-of-the-art methods.

Hyperspectral Denoising Using Unsupervised Disentangled Spatio-Spectral Deep Priors

Image denoising is often empowered by accurate prior information. In recent years, data-driven neural network priors have shown promising performance for RGB natural image denoising. Compared to classic handcrafted priors (e.g., sparsity and total variation), the "deep priors" are learned using a large number of training samples -- which can accurately model the complex image generating process. However, data-driven priors are hard to acquire for hyperspectral images (HSIs) due to the lack of training data. A remedy is to use the so-called unsupervised deep image prior (DIP). Under the unsupervised DIP framework, it is hypothesized and empirically demonstrated that proper neural network structures are reasonable priors of certain types of images, and the network weights can be learned without training data. Nonetheless, the most effective unsupervised DIP structures were proposed for natural images instead of HSIs. The performance of unsupervised DIP-based HSI denoising is limited by a couple of serious challenges, namely, network structure design and network complexity. This work puts forth an unsupervised DIP framework that is based on the classic spatio-spectral decomposition of HSIs. Utilizing the so-called linear mixture model of HSIs, two types of unsupervised DIPs, i.e., U-Net-like network and fully-connected networks, are employed to model the abundance maps and endmembers contained in the HSIs, respectively. This way, empirically validated unsupervised DIP structures for natural images can be easily incorporated for HSI denoising. Besides, the decomposition also substantially reduces network complexity. An efficient alternating optimization algorithm is proposed to handle the formulated denoising problem. Semi-real and real data experiments are employed to showcase the effectiveness of the proposed approach.